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10.5 The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression

This section focuses on the entity fixed effects model and presents model assumptions that need to hold in order for OLS to produce unbiased estimates that are normally distributed in large samples. These assumptions are an extension of the assumptions made for the multiple regression model (see Key Concept 6.4) and are given in Key Concept 10.3. We also briefly discuss standard errors in fixed effects models which differ from standard errors in multiple regression as the regression error can exhibit serial correlation in panel models.

Key Concept 10.3

The Fixed Effects Regression Assumptions

In the fixed effects model \[ Y_{it} = \beta_1 X_{it} + \alpha_i + u_{it} \ \ , \ \ i=1,\dots,n, \ t=1,\dots,T, \] we assume the following:

  1. The error term \(u_{it}\) has conditional mean zero, that is, \(E(u_{it}|X_{i1}, X_{i2},\dots, X_{iT} = 0)\).

  2. \((X_{i1}, X_{i2}, \dots, X_{it}, u_{i1}, \dots, u_{iT})\), \(i=1,\dots,n\) are i.i.d. draws from their joint distribution.

  3. Large outliers are unlikely, i.e., \((X_{it}, u_{it})\) have nonzero finite fourth moments.

  4. There is no perfect multicollinearity.

When there are multiple regressors, \(X_{it}\) is replaced by \(X_{1,it}, X_{2,it}, \dots, X_{k,it}\).

The first assumption is that the error is uncorrelated with all observations of the variable \(X\) for the entity \(i\) over time. If this assumption is violated, we face omitted variables bias. The second assumption ensures that variables are i.i.d. across entities \(i=1,\dots,n\). This does not require the observations to be uncorrelated within an entity. The \(X_{it}\) are allowed to be autocorrelated within entities. This is a common property of time series data. The same is allowed for errors \(u_{it}\). Consult Chapter 10.5 (Stock and Watson) of the book for a detailed explanation for why autocorrelation is plausible in panel applications. The second assumption is justified if the entities are selected by simple random sampling. The third and fourth assumptions are analogous to the multiple regression assumptions made in Key Concept 6.4.

Standard Errors for Fixed Effects Regression

Similar as for heteroskedasticity, autocorrelation invalidates the usual standard error formulas as well as heteroskedasticity-robust standard errors since these are derived under the assumption that there is no autocorrelation. When there is both heteroskedasticity and autocorrelation, the so-called heteroskedasticity and autocorrelation-consistent (HAC) standard errors need to be used. Clustered standard errors belong to these type of standard errors. They allow for heteroskedasticity and autocorrelated errors within an entity but not correlation across entities.

As shown in the examples throughout this chapter, it is fairly easy to specify usage of clustered standard errors in regression summaries produced by functions like coeftest() in conjunction with vcovHC() from the package sandwich. Conveniently, vcovHC() recognizes panel model objects (objects of class plm) and computes clustered standard errors by default.

The regressions conducted in this chapter are good examples for why usage of clustered standard errors is crucial in empirical applications of fixed effects models. For example, consider the entity and time fixed effects model for fatalities. Since fatal_tefe_lm_mod is an object of class lm, coeftest() does not compute clustered standard errors but uses robust standard errors that are only valid in the absence of autocorrelated errors.

# check class of the model object
#> [1] "lm"

# obtain a summary based on heteroskedasticity-robust standard errors 
# (no adjustment for heteroskedasticity only)
coeftest(fatal_tefe_lm_mod, vcov = vcovHC, type = "HC1")[1, ]
#>   Estimate Std. Error    t value   Pr(>|t|) 
#> -0.6399800  0.2547149 -2.5125346  0.0125470

# check class of the (plm) model object
#> [1] "plm"        "panelmodel"

# obtain a summary based on clustered standard errors 
# (adjustment for autocorrelation + heteroskedasticity)
coeftest(fatal_tefe_mod, vcov = vcovHC, type = "HC1")
#> t test of coefficients:
#>         Estimate Std. Error t value Pr(>|t|)  
#> beertax -0.63998    0.35015 -1.8277  0.06865 .
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The outcomes differ rather strongly: imposing no autocorrelation we obtain a standard error of \(0.25\) which implies significance of \(\hat\beta_1\), the coefficient on \(BeerTax\) at the level of \(5\%\). On the contrary, using the clustered standard error \(0.35\) results in a failure to reject the null hypothesis \(H_0: \beta_1 = 0\) at the same level, see equation (10.8). Consult Appendix 10.2 of the book (Stock and Watson) for insights on the computation of clustered standard errors.